Algorithm 1
Framework of ADMM
(1) Initialization: raw input
, Lagrange multiplier
, penalty parameter
|
(2) Alternating iteration:
|
(3) Stop, or set
|
Algorithm 2
ADMM for Hybrid High-Order Nonlocal Gradient Sparsity Regularization
(1) Initialization: raw input
, parameters
,
,
, Maxter,
|
(2) While (
) Do alternating iteration.
|
(3) Stop, or set
|
Algorithm 3
Spatially Adaptive HHONGS for Poissonian Image Deconvolution
Initialization: observed image
, convolution factor
, step size
, parameter
, squared domain size
|
(1) Solve the optimization problem based on Algorithm 2
|
(2) Based on current estimated image
, update parameter
|
(3) Stop, or set
|
Table 1.
Sensitivity Analysis of Parameter
Image |
|
Min (dB) | Max (dB) | Difference (dB) |
Mandril | 32.2481 | 32.6367 | 0.3886 |
Barbara | 28.5364 | 28.7567 | 0.2203 |
Cameraman | 30.0567 | 30.4891 | 0.4324 |
House | 29.8436 | 30.0406 | 0.1970 |
Peppers | 31.5478 | 31.8173 | 0.2695 |
Car | 27.2522 | 27.6345 | 0.3823 |
Dog | 32.3983 | 32.5169 | 0.1186 |
Cell | 25.3725 | 25.6724 | 0.2999 |
Moon | 27.3054 | 27.5242 | 0.2188 |
Organ | 29.7354 | 30.0125 | 0.2711 |
Table 2.
PSNR (dB) Comparison Based on the TV, HOTV, NGS, and HHONGS Methods
Kernel Size | Motion Blur | Gaussian Blur | Average Blur |
Method | TV | HOTV | NGS | HHONGS | TV | HOTV | NGS | HHONGS | TV | HOTV | NGS | HHONGS |
Mandril | 30.83 | 32.16 | 33.02 |
33.51
| 29.32 | 31.02 |
31.12
| 30.90 | 32.12 | 33.84 | 32.86 |
34.46
|
Barbara | 24.67 | 26.03 | 25.67 |
28.19
| 25.61 | 27.34 | 27.65 |
28.05
| 32.15 |
34.53
| 26.24 | 28.61 |
Cameraman | 30.78 |
32.11
| 31.18 | 31.83 | 26.72 | 29.02 | 28.22 |
30.95
| 33.52 | 36.14 | 35.94 |
36.85
|
House | 28.94 |
32.19
| 29.45 | 32.04 | 28.38 | 29.04 | 29.32 |
29.43
| 29.67 | 30.61 | 30.59 |
31.12
|
Pepper | 29.16 | 30.16 | 30.88 |
31.16
| 31.03 | 32.15 |
32.72
| 32.04 | 30.18 | 30.42 | 32.46 |
32.54
|
Car | 28.19 | 29.26 | 28.54 |
29.80
| 25.32 | 26.54 | 26.15 |
26.89
| 24.58 |
27.69
| 27.15 | 26.14 |
Dog | 29.90 | 31.13 | 29.19 |
31.36
| 30.94 | 32.11 | 31.28 |
32.51
| 29.86 | 31.18 | 33.47 |
34.67
|
Cell | 22.94 | 23.58 | 24.11 |
24.32
| 22.08 | 24.58 | 24.12 |
25.67
| 26.04 | 26.37 | 26.98 |
28.35
|
Moon | 24.96 | 25.69 | 24.63 |
26.35
| 23.57 | 24.38 | 26.14 |
27.52
| 25.69 | 27.16 | 27.35 |
29.99
|
Organ | 25.33 | 26.89 | 26.17 |
29.38
| 26.38 | 29.54 | 29.34 |
30.01
| 28.67 | 30.38 | 31.95 |
33.63
|
Average | 27.57 | 28.92 | 28.28 |
29.79
| 26.94 | 28.57 | 28.61 |
29.40
| 29.25 | 30.83 | 30.50 |
31.64
|
Table 3.
SSIM (%) Comparison Based on TV, HOTV, NGS, and HHONGS Methods
Kernel Size | Motion Blur | Gaussian Blur | Average Blur |
Method | TV | HOTV | NGS | HHONGS | TV | HOTV | NGS | HHONGS | TV | HOTV | NGS | HHONGS |
Mandril | 84.43 |
88.64
| 88.46 | 87.64 | 85.67 | 88.16 |
89.65
| 89.26 | 86.25 | 87.34 | 88.48 |
89.11
|
Barbara | 87.64 | 85.39 | 89.24 |
90.87
| 86.37 |
88.94
| 88.12 | 88.30 | 88.26 |
90.27
| 88.68 | 89.07 |
Cameraman | 84.64 | 90.01 | 89.37 |
90.24
| 87.64 | 86.34 | 89.13 |
89.54
| 89.35 | 90.57 |
91.68
| 88.28 |
House | 86.42 | 86.24 | 84.21 |
88.34
| 85.13 | 86.14 | 84.62 |
86.61
| 88.34 | 86.61 | 88.39 |
88.53
|
Pepper | 82.34 | 83.46 | 84.33 |
86.48
| 86.12 | 87.34 |
89.63
| 86.35 | 89.34 | 91.61 | 90.44 |
91.89
|
Car | 85.24 | 90.36 | 91.23 |
92.64
| 84.25 | 85.14 | 87.53 |
88.67
| 88.64 | 90.56 | 90.47 |
91.46
|
Dog | 87.72 | 89.37 | 89.43 |
90.72
| 83.14 | 84.35 | 83.35 |
85.13
| 90.42 | 90.63 | 90.54 |
91.22
|
Cell | 85.65 | 86.38 | 86.98 |
88.02
| 84.96 | 86.35 | 85.83 |
86.96
| 87.69 | 89.35 | 89.34 |
90.69
|
Moon | 84.33 | 85.69 | 86.66 |
87.39
| 86.39 | 87.93 | 87.37 |
89.32
| 88.36 | 89.78 | 90.36 |
91.22
|
Organ | 87.92 | 88.96 | 88.31 |
89.63
| 86.17 | 87.39 | 87.69 |
89.67
| 87.38 | 88.61 | 89.46 |
90.96
|
Average | 85.63 | 87.45 | 87.82 |
89.20
| 85.58 | 86.81 | 87.29 |
87.98
| 88.40 | 89.53 | 89.78 |
90.24
|
Table 4.
MSE Comparison Based on TV, HOTV, NGS, and HHONGS Methods
Kernel Size | Motion Blur | Gaussian Blur | Average Blur |
Method | TV | HOTV | NGS | HHONGS | TV | HOTV | NGS | HHONGS | TV | HOTV | NGS | HHONGS |
Mandril | 54.09 | 39.72 | 32.48 |
29.11
| 76.18 | 61.72 | 58.42 |
52.88
| 40.02 | 27.03 | 33.84 |
23.36
|
Barbara | 221.93 | 162.32 | 176.75 |
99.38
| 179.54 | 120.58 | 112.32 |
102.06
| 99.83 | 83.04 | 154.65 |
82.07
|
Cameraman | 54.76 | 60.26 | 49.81 |
42.70
| 139.21 | 81.60 | 98.16 |
52.46
| 28.95 | 15.88 | 16.63 |
13.47
|
House | 83.15 | 69.50 | 74.09 |
40.67
| 94.87 | 81.74 | 76.53 |
74.47
| 70.42 | 56.85 | 57.13 |
50.36
|
Pepper | 79.01 | 63.09 | 53.35 |
49.98
| 51.66 | 39.93 |
34.85
| 40.83 | 62.70 | 59.10 | 36.93 |
36.45
|
Car | 98.75 | 77.63 | 91.67 |
68.36
| 192.32 | 145.18 | 158.80 |
133.80
| 226.61 |
111.38
| 125.36 | 159.25 |
Dog | 67.02 | 50.18 | 78.83 |
47.79
| 52.39 | 40.11 | 48.67 |
36.62
| 67.22 | 49.86 | 29.38 |
22.31
|
Cell | 331.99 | 286.10 | 254.13 |
240.67
| 402.93 | 226.98 | 252.55 |
176.91
| 162.41 | 150.97 | 131.25 |
95.59
|
Moon | 208.74 | 176.04 | 225.35 |
151.43
| 286.76 | 237.84 | 158.43 |
115.27
| 175.94 | 125.81 | 120.39 |
65.47
|
Organ | 190.88 | 133.96 | 158.00 |
75.53
| 150.76 | 72.60 | 76.24 |
64.98
| 88.61 | 59.84 | 41.70 |
28.30
|
Average | 139.03 | 111.88 | 119.45 |
84.56
| 162.66 | 110.83 | 107.50 |
85.03
| 102.27 | 73.98 | 74.73 |
57.66
|
Table 5.
Computation Cost for TV, HOTV, NGS, and HHONGS (Image, Cameraman; Gaussian Kernel,
with Variance 1.5; Poisson Noise)
Method | TV
| HOTV
| NGS
| HHONGS
|
CPU time/s | 1.42 | 11.05 | 4628.37 | 1624.77 |
Regularization |
|
|
|
|
Solver | ALM | ADMM | ALM | ADMM |
Time for searching similar blocks/s | 0 | 0 | 4627.36 | 1621.93 |
Time for deblurring/s | 1.42 | 11.05 | 1.01 | 2.84 |
Number of iterations | 5 | 23 | 4 | 12 |
Time per iteration/s | 0.284 | 0.48 | 1157.09 | 0.24 |